SECTION 9.2 Polar Equations and Graphs 623 Other Equations Name Cardioid Limaçon without inner loop Limaçon with inner loop Polar equations θ = ± > a a a r cos , 0 θ = ± > > a b a b r cos , 0 θ = ± > > a b b a r cos , 0 θ = ± > a a a r sin , 0 θ = ± > > a b a b r sin , 0 θ = ± > > a b b a r sin , 0 Typical graph y x y x y x Name Lemniscate Rose with three petals Rose with four petals Polar equations θ ( ) = ≠ a a r cos 2 , 0 2 2 θ ( ) = > a a r sin 3 , 0 θ ( ) = > a a r sin 2 , 0 θ ( ) = ≠ a a r sin 2 , 0 2 2 θ ( ) = > a a r cos 3 , 0 θ ( ) = > a a r cos 2 , 0 Typical graph y x y x y x Table 7 (Continued) Sketching Quickly If a polar equation involves only a sine (or cosine) function, you can quickly obtain its graph by making use of Table 7, periodicity, and a short table. Sketching the Graph of a Polar Equation Quickly Graph the equation: θ = + r 2 2 sin EXAMPLE 14 Solution Because a b 2, = = the graph of this polar equation is a cardioid.The period of sinθ is 2 ,π so form a table using 0 2 , θ π ≤ ≤ compute r, plot the points r, , θ ( ) and sketch the graph of a cardioid as θ varies from 0 to 2 .π See Table 8 and Figure 37. Figure 37 θ = + r 2 2sin x u 5 0 u 5 p u 5 p– 2 u 5 3p –– 2 u 5 7p –– 4 u 5 p– 4 u 5 3p –– 4 u 5 5p –– 4 y 2 3 4 5 (2, p) (2, 0) 1 3p –– 2 ( ) 0, p–– 2 ( ) 4, θ θ = + r 2 2sin 0 + ⋅ = 2 2 0 2 π 2 + ⋅ = 2 2 1 4 π + ⋅ = 2 2 0 2 π3 2 ( ) + − = 2 2 1 0 π2 + ⋅ = 2 2 0 2 Table 8 Calculus Comment For those of you planning to study calculus, a comment about one important role of polar equations is in order. In rectangular coordinates, the equation x y 1, 2 2 + = whose graph is the unit circle, is not the graph of a function. In fact, it requires two functions to obtain the graph of the unit circle: y x 1 1 2 = − Upper semicircle y x 1 2 2 = − − Lower semicircle
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